Answer :
To find the monthly interest rate from an annual percentage yield (APY) of 4%, we need to understand the relationship between APY and the monthly interest rate.
Annual Percentage Yield (APY) is the effective annual rate of return taking into account the effect of compounding interest. The formula to convert APY to the monthly interest rate is derived from the concept of compound interest.
Firstly, the formula for converting APY to the monthly interest rate is:
[tex]\[ (1 + r_\text{monthly})^{12} = 1 + \text{APY} \][/tex]
Here, [tex]\( r_\text{monthly} \)[/tex] is the monthly interest rate as a decimal, and APY is the annual percentage yield as a decimal. Solving for the monthly interest rate, we get:
[tex]\[ r_\text{monthly} = (1 + \text{APY})^{\frac{1}{12}} - 1 \][/tex]
Given that the APY is 4%, which is [tex]\( \text{APY} = 0.04 \)[/tex] in decimal form, substituting this value into the formula, we calculate:
[tex]\[ r_\text{monthly} = (1 + 0.04)^{\frac{1}{12}} - 1 \][/tex]
After computing the above expression, we find that the monthly interest rate [tex]\( r_\text{monthly} \)[/tex] is approximately:
[tex]\[ r_\text{monthly} \approx 0.0032737397821989145 \][/tex]
To express this monthly interest rate as a percentage, we multiply by 100:
[tex]\[ r_\text{monthly\_percentage} = 0.0032737397821989145 \times 100 = 0.32737397821989145\% \][/tex]
Thus, the monthly interest rate corresponding to an APY of 4% is approximately 0.3274%, which, when rounded appropriately, matches the given options.
Therefore, the correct answer is:
A) [tex]\( 0.333 \% \)[/tex]
Annual Percentage Yield (APY) is the effective annual rate of return taking into account the effect of compounding interest. The formula to convert APY to the monthly interest rate is derived from the concept of compound interest.
Firstly, the formula for converting APY to the monthly interest rate is:
[tex]\[ (1 + r_\text{monthly})^{12} = 1 + \text{APY} \][/tex]
Here, [tex]\( r_\text{monthly} \)[/tex] is the monthly interest rate as a decimal, and APY is the annual percentage yield as a decimal. Solving for the monthly interest rate, we get:
[tex]\[ r_\text{monthly} = (1 + \text{APY})^{\frac{1}{12}} - 1 \][/tex]
Given that the APY is 4%, which is [tex]\( \text{APY} = 0.04 \)[/tex] in decimal form, substituting this value into the formula, we calculate:
[tex]\[ r_\text{monthly} = (1 + 0.04)^{\frac{1}{12}} - 1 \][/tex]
After computing the above expression, we find that the monthly interest rate [tex]\( r_\text{monthly} \)[/tex] is approximately:
[tex]\[ r_\text{monthly} \approx 0.0032737397821989145 \][/tex]
To express this monthly interest rate as a percentage, we multiply by 100:
[tex]\[ r_\text{monthly\_percentage} = 0.0032737397821989145 \times 100 = 0.32737397821989145\% \][/tex]
Thus, the monthly interest rate corresponding to an APY of 4% is approximately 0.3274%, which, when rounded appropriately, matches the given options.
Therefore, the correct answer is:
A) [tex]\( 0.333 \% \)[/tex]
